# Math Help - densities -- confusing topic for me

1. ## densities -- confusing topic for me

Random variables X and Y have joint density:
f_x,y(x,y) = c(y^(2)-x^(2))e^(-y) when -y<x<y, y>0
f_x,y(x,y) = 0 otherwise
Here, c is a constant.

I have shown that Y has a gamma density, and have found that c=1/8.

Now, how do I find the density of Y^(3)?
Any help is greatly appreciated! Thanks in advance.

2. we can find the density of Y by integrating the joint density with respect to x
thus getting
fy(Y) = (1/6)*(y^3)*e^(-y)

now we are interested in finding the pdf of z = g(y) = y^3, g(y) is a monotonic function of y thus the resulting density function is given by:

fz(Z) = abs(1 / g'(g-1(z)) )*fy(g-1(z))

where g-1 is the inverse of g in our case g-1(z) = z^(1/3), thus we get:

fz(Z) = [(1/3)*Z^(-2/3)]*(1/6)*Z*e^(-Z^(1/3)) =
[(1/18)*Z^(1/3)]*e^(-Z^(1/3)) for 0 <= Z <= inf